Consistent amalgamation for þ-forking (Q386624)

In simple theories, forking independence satisfies the ``independence theorem'' which states that if \(p(x,B)\) and \(q(x,C)\) are nonforking extensions of the same complete type over some model \(M\) contained in \(B\cap C\) for which \(B\) and \(C\) are forking-independent over M, then \(p(x,B)\cup q(x,C)\) does not fork over \(M\). (The authors of this paper refer to the conclusion of the independence theorem as ``independent amalgmation.'') Conversely, \textit{B. Kim} [J. Lond. Math. Soc., II. Ser. 57, No. 2, (1998; Zbl 0922.03048)] proved that any theory admitting a ``reasonable'' abstract notion of independence which satisfies the independence theorem (where forking independence is replaced by the abstract notion) must be simple and that the abstract notion of independence must actually be forking independence.NEWLINENEWLINERosy theories are a generalization of simple theories in which a particular notion of independence, namely \textit{thorn-independence}, is well-behaved. \textit{A. Onshuus} [J. Symb. Log. 71, No. 1, 1--21 (2006; Zbl 1103.03036)] proved that thorn-independence satisfies ``weak amalgamation'' in rosy theories in the sense that the independence theorem for thorn-independence is true if one is allowed to replace \(q(x,C)\) by an \(A\)-conjugate. Since there are rosy theories that are not simple (such as o-minimal theories), one cannot hope for the indpendence theorem for thorn-independence to be true.NEWLINENEWLINE That being said, thorn-independence in o-minimal theories does satisfy ``consistent amalgamation'' in the sense that once \(p(x,B)\cup q(x,C)\) is consistent, then it does not thorn-fork over \(M\). In this article, the authors prove two results concerning consistent amalgmation in rosy theories. The bulk of the paper is devoted to giving an example of a rosy theory for which consistent amalgamation fails; the example is a modified Hrushovski construction. In a positive direction, the authors prove that in a rosy \textit{dependent} theory, consistent amalgamation holds under an additional technical assumption. More precisely, if \(p(x,a)\) and \(q(x,b)\) are non-thorn-forking extensions of a complete type over a common model \(M\) such that \(p(x,a)\cup q(x,b)\) is consistent and such that \(a\) and \(b\) start an infinite thorn-independent indiscernible sequence, then \(p(x,a)\cup q(x,b)\) does not thorn-fork over \(M\). The authors point out that they do not know whether or not this technical assumption is necessary to obtain consistent amalgamation in rosy dependent theories.